Morality: Theory and Practice

Ought implies can.

-Kant's Law

Whether something is doable is irrelevant when it comes to determining whether it is right.

A separate question is what should we do, which is different from what is right. We should definitely do the most right thing we possibly can, but just because we can't do something does not mean that it is any less right.

A real example: There's nothing we can realistically do to stop much of the suffering undergone by wild animals through the predatory instinct. Yet the suffering of prey is very real and has ethical implications. Here we see something which has moral standing even though there appears to be nothing we can do to help the situation (beyond some trivial amount).

Ah, I see. I'm pretty sure you've run up against the "ought implies can" issue, not the issue of demandingness. IIRC, this is a contested principle, but I don't really know much about it other than Kant originally endorsing it. I think the first part of Larks' answer gives you a good idea of what consequentialists would say in response to this issue.

Do we want to promote a theory that says "the very best thing is right, everything else is wrong",

No. That just means the better your imagination gets, the less you do.

Consequentialism solves all of this:

1. Give each possible world a "goodness" or "awesomeness" or "rightness" number (utility)
2. Figure out the probability distribution over possible outcomes of each action you could take.
3. Choose the action that has highest mean awesomeness.

If something is impossible, it won't be reachable from the action set and therefore won't come into it. If something is bad, but nothing you can do will change it, it will cancel out. If some outcome is not actually preferable to some other outcome, you will have marked it as such in your utility assignment. If something good also comes with something worse, the utility of that possibility should reflect that. Etcetera.

In practice, you don't actually compute this, because it is uncomputable. Instead you follow simple rules that get you good results, like "don't throw away money" and "don't kill people" and "feed yourself" (Notice how the rules are justified by appealing to their expected consequences, though).

Consequentialism is a method for choosing an action from the set of possible actions. If "the best action is impossible" it shouldn't have been in the option set in the first place.

However, I think you might like to look into scalar consequentialism.

Plain Scalar Consequentialism: Of any two things a person might do at any given moment, one is better than another to the extent that the its overall consequences are better than the other’s overall consequences.

If you figured out the theoretically correct action, you wouldn't need to approximate it. I mean figure out the theoretically correct moral theory, then approximate it to the best of your ability. You're not approximating the output of an algorithm, you're approximating an algorithm (e.g. because the correct algorithm requires too much data, or time, or rationality...).